Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential

We investigate the asymptotic behavior, as t   goes to infinity, for a semilinear hyperbolic equation with asymptotically small dissipation and convex potential. We prove that if the damping term behaves like Ktα as t→+∞t→+∞, for some K>0K>0 and α∈]0,1[α∈]0,1[, then every global solution converges weakly to an equilibrium point. This result is a positive answer to a question left open in the paper of Cabot and Frankel (2012) [6].